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National and international surveys highlight a strong disparity in males and females results in mathematics: boys outperform female in almost all the countries and in all school levels. In the international context, Italy appears to be one of the countries in which the results related to gender differences in mathematics are more alarming and INVALSI tests’ data confirm a very marked gender gap in favour of males at all school levels. In this paper we present the international situation regarding gender differences in mathematics starting from large scale assessment and paying particular attention to the Italian situation. Hence we present a detailed overview of the causes underlying the gender gap in mathematics, analysing the main studies on this issue and trying to provide a complete, though complex, interpretation of this phenomenon.

In this thesis, we present three topics concerning the study of Hilbert function. In the first topic, we obtained a partial solution to Froberg's conjecture. In the second topic, we gave the bounds for the second Hilbert coefficients of filtered modules. In the third part, we study the preservation of Hilbert function under small perturbations.

Several researches in STEM education research highlight the advantages of an inte- grated approach to these disciplines that relates knowledge and know-how, design and implementation, theoretical and practical problems [5, 4, 6]. In some researches, the effectiveness of these approaches on students conceptual understanding and motivation and has been studied through the use of quantitative analysis tools such as cluster analysis (CLA) [1, 7]. Through CLA it is possible to characterize students analyzing the strategies they deploy to tackle, for example, questionnaires built so as to investigate the lines of reasoning implemented by them when they are proposed with problematic situations. In particular, it is possible to characterize the students in terms of a limited number, m, of typical ways of answering the questionnaire questions [2]. Each student is therefore identified by a binary vector (each component can be 1 or 0) with m dimensions. Cluster Analysis techniques allow students to be grouped into homogeneous groups based on com- mon characteristics and the representation of these groups is ideally referred to an m-sized space. However, for reasons of simplicity and clarity, it is often preferred to perform the representation of groups in three or two dimensional spaces. One of the techniques used for this purpose is Multidimensional Scaling [3] (MDS). It allows the researcher to move from the m-sized space to a space with a smaller number of dimensions that is a function of the initial m-dimensional representation, preserving the global distances between the group elements. However, the application of the MDS methodologies depends strongly on the typology of the initial data and a non-thorough knowledge of the mathematical details at their base can lead to obtaining results that are not reliable and / or of little significance. In this paper we will study a MDS methodology based on Principal Component Analysis, with particular reference to a set of binary data, highlighting how the results obtained through this methodology can be reliable and significant for the researcher in education.

National and international large-scale assessments of mathematics show that, in most nations, males achieve better results than females and Italy is one of the countries with a greater gap. Many research studies in mathematics education have analysed this issue, using both quantitative and qualitative methods to understand the sources and characteristics of this gap. This study focuses on a specific Grade 10 task that requires algebraic manipulations of powers with the same base. Item-level analysis enables the study of gender differences on specific content, before using the lenses of mathematics education theories to interpret macro-phenomena emergeing from standardized assessment results. The quantitative analysis, carried out using the Rasch statistical model, highlights a gender gap in favour of males in this task and, furthermore, a variance in choices of incorrect options between males and females; the interviews conducted provide a key to understanding this phenomenon in terms of didactic contract.

Mathematical music theory helps us investigate musical compositions in mathematical terms. Some hints can be extended towards the visual arts. Mathematical approaches can also help formalize a "translation" from the visual domain to the auditory one and vice versa. Thus, a visual artwork can be mathematically investigated, then translated into music. The final, refined musical rendition can be compared to the initial visual idea. Can an artistic idea be preserved through these changes of media? Can a non-trivial pattern be envisaged in an artwork, and then still be identified after the change of medium? Here, we consider a contemporary installation and an ensemble musical piece derived from it. We first mathematically investigate the installation, finding its patterns and structure, and then we compare them with structure and patterns of the musical composition. In particular, we apply two concepts of mathematical music theory, the Quantum GestART and the gestural similarity conjecture, to the analysis of Qwalala, realized for the Venice Biennale by Pae White, comparing it to its musical rendition in the homonymous piece for harp and ensemble composed by Federico Favali. Some sketches of generalizations follow, with the "Souvenir Theorem" and the "Art Conjecture."

Trascurare o minimizzare il ruolo della Matematica di Euclide nella scuola secondaria di secondo grado può essere un grave errore: un impoverimento culturale per tutti, e un effettivo handicap" per chi si iscriverà ad un corso di laurea di carattere tecnico-scientifico.

Il paper mette a fuoco gli esiti dell’indagine statistica rivolta agli insegnanti di Ma- tematica di scuola Primaria del contesto italiano, circa l’adozione della Didattica a Distanza (DaD), impiegata durante il periodo di emergenza sanitaria da Covid-19. In una prima parte si inquadra la DaD generale da un punto di vista storico e teorico-pratico e, dopo, il focus diviene l’insegnamento e l’apprendimento della Matematica, attraverso l’e-learning. In una seconda parte viene presentata la ricerca e i risultati ad essa correlati. L’indagine statistica, messa in atto attraverso un questionario, qui interamente riportato, ha avuto l’obiettivo di esaminare le rela- zioni intercorrenti tra molteplici variabili che sono risultate essere rilevanti nel processo educa- tivo-didattico della Matematica. Dallo studio sono emerse molteplici problematiche didattiche, pedagogiche ed epistemologiche legate all’implementazione della DaD, all’“impreparazione” degli insegnanti in riferimento alla DaD e, in generale, all’uso delle tecnologie dal punto di vi- sta didattico. In ultimo, emerge una forte esigenza, percepita dagli insegnanti stessi, di forma- zione sui temi qui indagati e relativa ad una seria implementazione ad hoc delle metodologie e- learning nel contesto scolastico della scuola Primaria, metodologie queste ancora poco note nel panorama italiano.

In this paper, we examine the evolution of a specific mathematical problem, i.e. the nine-point conic, a generalisation of the nine-point circle due to Steiner. We will follow this evolution from Steiner to the Neapolitan school (Trudi and Battaglini) and finally to the contribution of Beltrami that closed this journey, at least from a mathematical point of view (scholars of elementary geometry, in fact, will continue to resume the problem from the second half of the 19th to the beginning of the 20th century). We believe that such evolution may indicate the steady development of the mathematical methods from Euclidean metric to projective, and finally, with Beltrami, with the use of quadratic transformations. In this sense, the work of Beltrami appears similar to the recent (after the anticipations of Magnus and Steiner) results of Schiaparelli and Cremona. Moreover, Beltrami's methods are closely related to the study of birational transformations, which in the same period were becoming one of the main topics of algebraic geometry. Finally, our work emphasises the role played by the nine-point conic problem in the studies of young Beltrami who, under Cremona's guidance, was then developing his mathematical skills. To this end, we make considerable use of the unedited correspondence Beltrami – Cremona, preserved in the Istituto Mazziniano, Genoa.

Research, at both national and international levels, has started to focus on the gender gap regarding student performances in technological/scientific topics. Fervent ideological and epistemological debates have sprung up, particularly concerning the differences in mathematics test results between males and females (with males outshining females in most countries). In this paper, we analyse some questions taken from the INVALSI mathematics tests administered at varying educational stages; emergent didactic phenomena are investigated via one of the key concepts of mathematics education, the didactic contract. From a quantitative analysis, based on the percentage of answers given and Rasch Model results, it is possible to identify the presence of significant gender differences between the performances which can be traced back to the effects of the didactic contract. Furthermore, study of the features which accompany these gender gaps in the case studies allows us to investigate the gender gap in mathematics via a new interpretative method which also takes into account student ability levels.

The paper presents some reflections and activities developed by researchers and teachers involved in teacher education programs on cultural transposition. The construct of cultural transposition is presented as a condition for decentralizing the didactic practice of a specific cultural context through contact with other didactic practices of different cultural contexts. We discuss the background theoretical issues of this approach and also give an analysis of two examples of cultural transposition experiences carried out in Italy. In particular, by means of qualitative analysis of some excerpts, discussions, and interviews, we show that the contact with different perspectives coming from China and Russia fostered educational practices and reflections on whole number arithmetic education.

How can discrete pitches and chords emerge from the continuum of sound? Using a quantum cognition model of tonal music, we prove that the associated Schrödinger equation in Fourier space is invariant under continuous pitch transpositions. However, this symmetry is broken in the case of transpositions of chords, entailing a discrete cyclic group as transposition symmetry. Our research relates quantum mechanics with music and is consistent with music theory and seminal insights by Hermann von Helmholtz.

While end users can acquire full 3D gestures with many input devices, they often capture only 3D trajectories, which are 3D uni-path, uni-stroke single-point gestures performed in thin air. Such trajectories with their $(x,y,z)$ coordinates could be interpreted as three 2D stroke gestures projected on three planes,\ie, $XY$, $YZ$, and $ZX$, thus making them admissible for established 2D stroke gesture recognizers. To investigate whether 3D trajectories could be effectively and efficiently recognized, four 2D stroke gesture recognizers, \ie, \$P, \$P+, \$Q, and Rubine, are extended to the third dimension: $\$P^3$, $\$P+^3$, $\$Q^3$, and Rubine-Sheng, an extension of Rubine for 3D with more features. Two new variations are also introduced: $\F for flexible cloud matching and FreeHandUni for uni-path recognition. Rubine3D, another extension of Rubine for 3D which projects the 3D gesture on three orthogonal planes, is also included. These seven recognizers are compared against three challenging datasets containing 3D trajectories, \ie, SHREC2019 and 3DTCGS, in a user-independent scenario, and 3DMadLabSD with its four domains, in both user-dependent and user-independent scenarios, with varying number of templates and sampling. Individual recognition rates and execution times per dataset and aggregated ones on all datasets show a highly significant difference of $\$P+^3$ over its competitors. The potential effects of the dataset, the number of templates, and the sampling are also studied.

This paper presents a study aimed at investigating the didactic potentiality of the use of an artefact, useful to construct mathematical meanings concerning the coordination of different points of view, in the observation of a real object/toy. In our view, the process of meaning construction can be fostered by the use of adequate artefacts, but it requires a teaching/learning model, which explicitly takes care of the evolution of meanings, from those personal, emerging through the activities, to the mathematical ones, aims of the teaching intervention. The main hypothesis of this study is that the alternation between different semiotic systems, graphical system, verbal system and system of gestures, can determine the evolution of the learning objectives, that are the coordination of different points of view. The Theory of Semiotic Mediation offers the theoretical framework suitable to design the teaching sequence and to analyze the collected data. The study involved 15 Kindergarten students, (5 or 6 years old) in which we asked children to represent a Lego block and discuss drawings observed from different points of view.

This contribution is centred on a case study of collaborative teaching, carried out among Upper Secondary School Math teachers. Here we present the context, the forms of the implemented collaboration and the effects that derived from the collaborative teaching. The used investigation tool was the semi-structured interview aimed to induce teachers to gradually reflect on their self and their teaching modus operandi. From the qualitative analysis emerged that collaborative teaching integrated different ways of teaching; it has been a stimulus for a etching “revision” and the possibility to improve the didactic-educational practice. For all teachers this way to define the teaching activities was a valuable “tool” to see mathematics as part of a unified knowledge, creating bridges between scientific and humanistic disciplines, in terms of content and methods.

Cos’è la simmetria in musica? Come investigare suoni e strutture musicali alla luce della simmetria? E quale può essere, in questa ricerca, il ruolo della matematica? Il premio Nobel per la Fisica F. Wilczek definisce la simmetria come un “cambiamento senza cambiamento”, un’invarianza rispetto a una trasforma- zione. La ricerca di simmetria e di bellezza può essere intesa come la ricerca di equilibrio fra regolarità e variazione. Se la regolarità aiuta la memoria, è invece la varietà a stimolare l’attenzione e la curiosità. Argomenti fisico-ma- tematici come la teoria dei gruppi, l’acustica e la teoria dei segnali fornisco- no strumenti di indagine anche all’estetica musicale. Il tema della simmetria fra matematica e musica è il filo conduttore dei testi di fisici, matematici, teorici della musica e compositori presenti in questa antologia, scelti e tradotti da Federico Favali e Maria Mannone. Il volume comprende scritti di D. Kempf, G. Rochberg, J. W. Bernard, D. Wilson, R. Donnini, D. Tymoczko, V. Hart, O. Fernández Herrero, G. W. Don - K. K. Muir - G. B. Volk - J. S. Walker. Prefazione di David Fontanesi; postfazione di Giovanni Albini.

Concepts and formalism from acoustics are often used to exemplify quantum mechanics. Conversely, quantum mechanics could be used to achieve a new perspective on acoustics, as shown by Gabor studies. Here, we focus in particular on the study of human voice, considered as a probe to investigate the world of sounds. We present a theoretical framework that is based on observables of vocal production, and on some measurement apparati that can be used both for analysis and synthesis. In analogy to the description of spin states of a particle, the quantum-mechanical formalism is used to describe the relations between the fundamental states associated with phonetic labels such as phonation, turbulence, and supraglottal myoelastic vibrations. The intermingling of these states, and their temporal evolution, can still be interpreted in the Fourier/Gabor plane, and effective extractors can be implemented. The bases for a Quantum Vocal Theory of Sound, with implications in sound analysis and design, are presented., 32 pages, 11 figures

“Welcome in the Mile High City”, benvenuti nella città a un miglio di altitudine, è la scritta che accoglie i visitatori a Denver, nello stato del Colorado, nella profonda America delle Montagne Rocciose. Denver è nota agli amanti di sport ed esplorazioni, nonché agli appassionati di paleontologia, per i numerosi fossili di dinosauri. Ed è nota agli studiosi per il gigantesco Convention Center, scelto come sede per il Joint Mathematics Meetings dello scorso gennaio 2020 (JMM 2020), vedi Figura 1.

“The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as poetry” (Bertrand Russel, Mysticism and Logic, 1910). This sentence, quoted by François Le Lionnais in his work La Beauté en Mathématiques in [1], reflects his conception of a deep bond between mathematics and literature. He had a multifaceted education and was an erudite and founder of the Oulipo with Raymond Queneau. Even though he was neither a “professional” mathematician nor a “professional” man of letters but only an épicurien passionné as he defined himself [2],1 while alive, he channelled his interests in the theorisation of the so-called littérature potentielle (potential literature) “pour exciter les curieux d’insolite et faire réfléchir les passionnés de littérature aussi bien que le fanatiques de mathématiques” [3]. The purpose of this chapter is outline the figure of François Le Lionnais (who we will refer to from here on as FLL as he himself liked to do [4]) through the analysis of his most meaningfulworks,2 which, in the current literature, appear as subordinated to the works of the more famous and researched Queneau.

It is with great pleasure that we dedicate the special issue Functional Analytic Methods in Partial Differential Equations of Complex Variables and Elliptic Equations to the 60th birthday of Massim...

Musical gestures connect the symbolic layer of the score to the physical layer of sound. I focus here on the mathematical theory of musical gestures, and I propose its generalization to include braids and knots. In this way, it is possible to extend the formalism to cover more case studies, especially regarding conducting gestures. Moreover, recent developments involving comparisons and similarities between gestures of orchestral musicians can be contextualized in the frame of braided monoidal categories. Because knots and braids can be applied to both music and biology (they apply to knotted proteins, for example), I end the article with a new musical rendition of DNA., Comment: 23 pages, 16 figures